Nima Hamidi

Nima Hamidi, Mohsen Bayati

This paper studies the stochastic linear bandit problem, where a decision-maker chooses actions from possibly time-dependent sets of vectors in $\mathbb{R}^d$ and receives noisy rewards. The objective is to minimize regret, the difference between the cumulative expected reward of the decision-maker and that of an oracle with access to the expected reward of each action, over a sequence of $T$ decisions. Linear Thompson Sampling (LinTS) is a popular Bayesian heuristic, supported by theoretical analysis that shows its Bayesian regret is bounded by $\widetilde{\mathcal{O}}(d\sqrt{T})$, matching minimax lower bounds. However, previous studies demonstrate that the frequentist regret bound for LinTS is $\widetilde{\mathcal{O}}(d\sqrt{dT})$, which requires posterior variance inflation and is by a factor of $\sqrt{d}$ worse than the best optimism-based algorithms. We prove that this inflation is fundamental and that the frequentist bound of $\widetilde{\mathcal{O}}(d\sqrt{dT})$ is the best possible, by demonstrating a randomization bias phenomenon in LinTS that can cause linear regret without inflation.We propose a data-driven version of LinTS that adjusts posterior inflation using observed data, which can achieve minimax optimal frequentist regret, under additional conditions. Our analysis provides new insights into LinTS and settles an open problem in the field.

Mohsen Bayati, Nima Hamidi, Ramesh Johari et al.

We investigate a Bayesian $k$-armed bandit problem in the \emph{many-armed} regime, where $k \geq \sqrt{T}$ and $T$ represents the time horizon. Initially, and aligned with recent literature on many-armed bandit problems, we observe that subsampling plays a key role in designing optimal algorithms; the conventional UCB algorithm is sub-optimal, whereas a subsampled UCB (SS-UCB), which selects $Θ(\sqrt{T})$ arms for execution under the UCB framework, achieves rate-optimality. However, despite SS-UCB's theoretical promise of optimal regret, it empirically underperforms compared to a greedy algorithm that consistently chooses the empirically best arm. This observation extends to contextual settings through simulations with real-world data. Our findings suggest a new form of \emph{free exploration} beneficial to greedy algorithms in the many-armed context, fundamentally linked to a tail event concerning the prior distribution of arm rewards. This finding diverges from the notion of free exploration, which relates to covariate variation, as recently discussed in contextual bandit literature. Expanding upon these insights, we establish that the subsampled greedy approach not only achieves rate-optimality for Bernoulli bandits within the many-armed regime but also attains sublinear regret across broader distributions. Collectively, our research indicates that in the many-armed regime, practitioners might find greater value in adopting greedy algorithms.

Nima Hamidi, Mohsen Bayati

Recent growing adoption of experimentation in practice has led to a surge of attention to multiarmed bandits as a technique to reduce the opportunity cost of online experiments. In this setting, a decision-maker sequentially chooses among a set of given actions, observes their noisy rewards, and aims to maximize her cumulative expected reward (or minimize regret) over a horizon of length $T$. In this paper, we introduce a general analysis framework and a family of algorithms for the stochastic linear bandit problem that includes well-known algorithms such as the optimism-in-the-face-of-uncertainty-linear-bandit (OFUL) and Thompson sampling (TS) as special cases. Our analysis technique bridges several streams of prior literature and yields a number of new results. First, our new notion of optimism in expectation gives rise to a new algorithm, called sieved greedy (SG) that reduces the overexploration problem in OFUL. SG utilizes the data to discard actions with relatively low uncertainty and then choosing one among the remaining actions greedily. In addition to proving that SG is theoretically rate optimal, our empirical simulations show that SG outperforms existing benchmarks such as greedy, OFUL, and TS. The second application of our general framework is (to the best of our knowledge) the first polylogarithmic (in $T$) regret bounds for OFUL and TS, under similar conditions as the ones by Goldenshluger and Zeevi (2013). Finally, we obtain sharper regret bounds for the $k$-armed contextual MABs by a factor of $\sqrt{k}$.